A generalized central sets theorem and applications

The central sets theorem originally proven by H. Furstenburg is a powerful result which is applicable to derive many combinatorial conclusions. Furstenburg's original theorem applied to NN and finitely many sequences in ZZ. Some strengthenings of this theorem have been derived first by V. Bergelson and N. Hindman in 1990. Later in 2008, D. De, N. Hindman, and D. Strauss proved a stronger version of the central sets theorem for arbitrary semigroups S which applied to all sequences in S. We provide here a generalization of the stronger version and some applications of this new generalization.